We observe that the above theorem is almost optimal in view of the results presented in [1]}. There, for any lattice \(S \subset \mathbb {R}^2\)  in the time-frequency plane, explicit examples \(f,g \in L^2(\mathbb {R})\)  were constructed which do not agree up to global phase but which satisfy that
\(\left|\mathcal {G}f (x,\omega ) \right| = \left|\mathcal {G}g (x,\omega ) \right|, \qquad (x,\omega ) \in S.\) 
